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Heteroskedasticity

by , PhD

Heteroskedasticity is the violation of the assumption, commonly made in linear regression models, that all the errors of the regression have the same variance.

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The assumption

Consider the linear regression[eq1]where $y_{i}$ is the regressand, $x_{i}$ are the regressors, $eta $ is the vector of regression coefficients, $\varepsilon _{i}$ is the error term and the observations are $i=1,\ldots ,N$.

Sometimes we make the assumption that all the error terms have the same variance, that is, [eq2]

When this assumption is met, we say that the errors are homoskedastic (or homoscedastic).

The violation

On the contrary, when the errors pertaining to different observations do not have the same variance, the assumption is violated and the errors are said to be heteroskedastic (or heteroscedastic). In this case, we also say that the regression suffers from heteroskedasticity.

Conditional heteroskedasticity

We often make an assumption stronger than homoskedasticity, called conditional homoskedasticity:[eq3]where X is the design matrix (i.e., the matrix whose $N$ rows are the vectors of regressors $x_{i}$ for $i=1,\ldots ,N$).

In other words, we assume that the variance of the errors is constant conditional on the design matrix.

When this assumption is violated, the regression is said to suffer from conditional heteroskedasticity.

Consequence

Conditional homoskedasticity is one of the assumptions of the Gauss-Markov theorem, which states that under certain conditions the OLS estimator is the best linear unbiased estimator (BLUE) of the vector of regression coefficients. Therefore, OLS is not guaranteed to be BLUE when a regression suffers from heteroskedasticity.

What to do: the WLS/GLS estimator

If all the conditions of the Gauss-Markov theorem except homoskedasticity are met, OLS is not the BLUE estimator, but the weighted least squares estimator (a special case of the generalized least squares estimator) is.

How to detect it

There are numerous statistical tests that can be used to detect heteroskedasticity, for example, the Goldfeld-Quandt, Breusch-Pagan and White tests. Most statistical packages have implementations of these test. For more information about these test, you can refer, for example, to Greene (2017) and Gurajati (2017).

More details

The mathematical details are discussed in the lecture on the generalized least squares estimator.

References

Greene, W.H. (2017) Econometric analysis, 8th edition, Pearson.

Gujarati, D.N. (2017) Basic econometrics, 5th edition, McGraw-Hill.

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